Inference for Who? Young adults. What? Heart rate (beats per minute). When? Where? In a physiology lab. How? Take pulse at wrist for one minute. Why? Part of an evaluation of general health.

Inference for What is the mean heart rate for all young adults? Use the sample mean heart rate, , to make inferences about the population mean heart rate, .

Inference for Sampling distribution of Shape: Approximately normal Center: Mean, Spread: Standard Deviation,

Problem The population standard deviation, is unknown. Therefore, is unknown as well.

Solution Use the sample standard deviation, s to get the standard error of

Problem The distribution of the standardized sample mean does not follow a normal model.

Solution The distribution of the standardized sample mean does follow a Student’s t-model with df = n – 1. A little history about Student’s t model. This model was developed by an individual working at the Guinness Brewery in Dublin Ireland.

Guinness is one of the oldest breweries in the world and has been brewing beer at the St. James’s Gate Brewery in Dublin since 1759. In the 1920 a brewer, who was also a statistician, named William Sealy Gosset was working on data on barley. He discovered that his inferences were not as good as was predicted by theory and he set about to find out why. The reason, Table Z was not appropriate when using the sample standard deviation. Gosset went on to develop Table T to overcome this problem. Because Guinness had a policy against publishing any results of research coming out of the brewery, Gosset had to publish his results under a pseudonym – A. Student.

Later, William Gosset went on to be the Master Brewer for Guinness.

Inference for Do NOT use Table Z! Use Table T instead! Table Z

Conditions Randomization condition. 10% condition. Nearly normal condition.

Randomization Condition Data arise from a random sample from some population. Data arise from a randomized experiment.

10% Condition The sample is no more than 10% of the population. Not as critical for means as it is for proportions.

Nearly Normal Condition The data come from a population whose shape is unimodal and symmetric. Look at the distribution of the sample. Could the sample have come from a normal model?

Confidence Interval for

Table T df 1 2 3 4 n–1 Confidence Levels 80% 90% 95% 98% 99%

Inference for What is the mean heart rate for all young adults? Use the sample mean heart rate, , to make inferences about the population mean heart rate, .

Sample Data Random sample of n = 25 young adults. Heart rate – beats per minute 70, 74, 75, 78, 74, 64, 70, 78, 81, 73 82, 75, 71, 79, 73, 79, 85, 79, 71, 65 70, 69, 76, 77, 66

Summary of Data n = 25 = 74.16 beats s = 5.375 beats = 1.075 beats

Conditions Randomization condition: 10% condition: Met because we have a random sample of 25 young adults. 10% condition: Met because 25 is less than 10% of all young adults that could have been sampled.

Conditions Nearly Normal Condition Could the sample have come from a population described by a normal model?

Heart rate

Nearly Normal Condition Normal quantile plot – data follows straight line for a normal model. Box plot – symmetric. Histogram – unimodal and symmetric.

Confidence Interval for

Table T df 1 2 3 4 24 2.064 Confidence Levels 80% 90% 95% 98% 99%

Confidence Interval for

71.94 beats/min and 76.38 beats/min Interpretation We are 95% confident that the population mean heart rate of young adults is between 71.94 beats/min and 76.38 beats/min

Interpretation Plausible values for the population mean. 95% of intervals produced using random samples will contain the population mean.

JMP:Analyze – Distribution Mean 74.16 Std Dev 5.375 Std Err Mean 1.075 Upper 95% Mean 76.38 Lower 95% Mean 71.94 N 25

Test of Hypothesis for Could the population mean heart rate of young adults be 70 beats per minute or is it something higher?

Test of Hypothesis for Step 1: State your null and alternative hypotheses.

Test of Hypothesis for Step 2: Check conditions. Randomization condition, met. 10% condition, met. Nearly normal condition, met.

Test of Hypothesis for Step 3: Calculate the test statistic and convert to a P-value.

Summary of Data n = 25 = 74.16 beats s = 5.375 beats = 1.075 beats

Value of Test Statistic Use Table T to find the P-value.

Table T The P-value is less than 0.005. 3.87 One tail probability 0.10 0.05 0.025 0.01 0.005 P-value df 1 2 3 4 24 2.064 2.492 2.797 3.87 The P-value is less than 0.005.

Test of Hypothesis for Step 4: Use the P-value to reach a decision. The P-value is very small, therefore we should reject the null hypothesis.

Test of Hypothesis for Step 5: State your conclusion within the context of the problem. The mean heart rate of all young adults is more than 70 beats per minute.

Alternatives

JMP:Analyze – Distribution Test Mean t-test Hypothesized value 70 Test statistic 3.87 Actual Estimate 74.16 Prob > |t| 0.0007 df 24 Prob > t 0.0004 Std Dev 5.375 Prob < t 0.9996

Example What is the mean alcohol content of beer? A random sample of 10 beers is taken and the alcohol content (%) is measured.

Sample Data – Alcohol (%) Molson Canadian 5.19 Heineken Dark 5.17 Michelob Dark 4.76 O’Keefe Canadian 4.96 Big Barrel Lager 4.32 Olympia Lager 4.78 Hamm’s 4.53 Miller Draft 4.85 Tsingtao 4.79 Guinness Stout 4.27

Test of Hypothesis for Step 1: State your null and alternative hypotheses.

Test of Hypothesis for Step 2: Check conditions. Randomization condition, met. 10% condition, met. Nearly normal condition, met.

Test of Hypothesis for Step 3: Calculate the test statistic and convert to a P-value.

Test of Hypothesis for Test statistic,

Table T The P-value is between 0.02 and 0.05. Two tail probability 0.20 0.10 0.05 P-value 0.02 df 1 2 3 4 9 2.262 2.397 2.821 The P-value is between 0.02 and 0.05.

Test of Hypothesis for Step 4: Use the P-value to reach a decision. The P-value is smaller than 0.05, therefore we should reject the null hypothesis.

Test of Hypothesis for Step 5: State your conclusion within the context of the problem. The population mean alcohol content of beer is not 5%.

JMP Output

Confidence Interval for

Interpretation We are 95% confident that the population mean alcohol content of beer is between 4.537% and 4.987%.

Interpretation The population mean alcohol content of beer could be any value between 4.537% and 4.987%. If we repeat the procedure that produces a confidence interval, 95% of intervals produced will capture the population mean.